Inverse method to calculate material properties using an insertion loss test

ABSTRACT

A method for calculating material properties of a material includes conducting two insertion loss tests of the material having a single thickness and a double thickness. These tests are conducted at a zero wavenumber. Utilizing these insertion loss tests, a dilatational wavespeed is computed. The method continues by calculating a shear wavespeed by performing three insertion loss tests of the material at single, double and triple thicknesses. These tests are conducted at a non-zero wavenumber. A shear wavespeed can be calculated from the dilatational wavespeed and these insertion loss tests. Lamé constants, Young&#39;s modulus, Poisson&#39;s ratio, and the shear modulus for the material of interest can then be calculated using the dilatational and shear wavespeeds.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefore.

CROSS REFERENCE TO OTHER RELATED APPLICATIONS

None.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The present invention relates to material properties measurement and,more particularly, to a method for measuring material properties usingwall displacement measurements recorded during an insertion lossexperiment.

(2) Description of the Prior Art

Insertion loss is a common measurement that is used to determine howeffective a piece of material attenuates acoustic energy at a specificfrequency. Insertion loss is calculated by projecting acoustic energy atpiece of material and measuring the pressure on the projector side andthe opposite side of the material, normally with hydrophones.

FIG. 1 depicts a typical setup for insertion loss. Sound pressure istransmitted to a test sample 10 by an acoustic projector 12. Acousticprojector 12 can transmit an acoustic wave at a preset frequency. Usinga 1 m by 1 m specimen, the minimum frequency is about 10 kHz. A firsthydrophone 14 is positioned on the opposite side of the sample 10 tomeasure the transmitted pressure. The ratio of the source pressure tothe transmitted pressure expressed in decibels is the insertion loss ofthe material. A second hydrophone 16 is positioned on the same side ofsample 10 as projector 12 to measure the reflected acoustic pressure.

Insertion loss can also be determined by measuring the motion of thetest sample with either accelerometers or laser velocimeters andcalculating the pressure field based on conservation of linear momentum.In the test setup shown here, a first laser velocimeter 18 is used tomeasure the acceleration and position of a first side 20 of sample 10. Asecond laser velocimeter 22 is used to measure the acceleration andposition of a second side 24 of sample 10. Laser velocimeters 18, 22 arepreferred because accelerometers must be positioned on sample 10 andmight interfere with the measurements. The projector 12 angle θ relativeto the test material can be changed so that the effects of acousticenergy at varying angles can be studied. Changing the excitation angle θis equivalent to changing the excitation wavenumber. Thus, the twoparameters that are typically varied during this test are frequency andwavenumber.

For underwater applications, the material is submerged in a fluid(normally water), and an underwater speaker or projector transmitsenergy at the material; however, a gaseous environment could be used.Because this test is only interested in acoustic attenuation of thematerial, the height and width of the test specimen are large comparedto its thickness. In view of this, the test specimen should have athickness between 10 mm and 100 mm. This prevents acoustic energy frommoving around the specimen and contaminating the transmitted pressurefield and interacting with the opposite side to the test specimen. Thetest is also dependent on the environment where it is conducted. Smalltest tanks prevent low frequency measurements due to reflection andreverberation of the acoustic energy. These are, however, practicallimitations and do not enter into this theoretical analysis.

SUMMARY OF THE INVENTION

One object of this invention is to accurately determine the materialproperties of a sample in an insertion loss experiment.

Another object of the present invention is to determine the materialproperties of dilatational wavespeed, shear wavespeed, Lamé constants,Young's modulus, and shear modulus of a material of interest.

The present invention features an inverse method where normal wallmovement measurements obtained during an insertion loss test arecombined to equal material properties. This allows for the calculationof Young's modulus, shear modulus, and Poisson's ratio from an insertionloss test. Alternatively, Lame constants and Poisson's ratio or complexdilatational and shear wavespeeds are also obtainable from this method.For dilatational wave energy, the test requires two material samples,one being twice as thick as the other. For shear wave energy, the testrequires three material samples, one being twice as thick as the firstand the second being three times as thick as the first. Measurements ofthese multiple samples allow the governing equations and test data becombined in a manner that results in an inverse method in which thematerial properties are closed form solutions of the measurement data.This is sometimes referred to as a linear inverse method.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features and advantages of the present invention will bebetter understood in view of the following description of the inventiontaken together with the drawings wherein:

FIG. 1 is a diagram of the test setup for the current invention;

FIG. 2 is a diagram showing the coordinate system used by the currentinvention;

FIGS. 3A and 3B are graphs of the transfer frequency magnitude and phaseangle at a zero degree excitation angle;

FIG. 4 is a graph of the function s versus frequency;

FIGS. 5A and 5B are graphs of the real and imaginary portions of theactual and estimated wavenumber alpha versus frequency;

FIGS. 6A and 6B are graphs of the real and imaginary portions of theactual and estimated dilatational wavespeed versus frequency;

FIGS. 7A and 7B are graphs of the transfer frequency magnitude and phaseangle at a fifteen degree excitation angle;

FIG. 8 is a graph of the function r and the angle of the discriminantversus frequency;

FIGS. 9A and 9B are graphs of the real and imaginary portions of theactual and estimated wavenumber beta versus frequency;

FIGS. 10A and 10B are graphs of the real and imaginary portions of theactual and estimated shear wavespeed versus frequency;

FIGS. 11A and 11B are graphs of the real and imaginary portions of theactual and estimated Lamé constant μ versus frequency;

FIGS. 12A and 12B are graphs of the real and imaginary portions of theactual and estimated Lamé constant λ versus frequency; and

FIGS. 13A and 13B are graphs of the real and imaginary portions of theactual and estimated Young's modulus versus frequency.

DETAILED DESCRIPTION OF THE INVENTION

The coordinate system of the test configuration is shown in FIG. 2.Projector 12 is oriented at an angle θ with respect to sample 10. Afirst measurement location 28 is located on the far side of sample 10from projector 12. This is the position where the beam from laservelocimeter 18 shown in FIG. 1 contacts surface 20. A second measurementlocation 26 corresponds to where second laser velocimeter 22 beamcontacts surface 24. Under the coordinate system, the z axis isorthogonal to the second surface of sample 10 with the origin at thissurface. Note that using this orientation results in b=0 and a having avalue less than zero (−h). The thickness of the sample, h, is a positivevalue. The y axis is oriented into the page.

The system model has three governing differential equations that arecoupled at their interfaces using conservation of linear momentum. Theacoustic pressure in the fluid on the projector side of the testspecimen is governed by the wave equation and is written in Cartesiancoordinates as [1]

$\begin{matrix}{{{\frac{\partial^{2}{p_{1}\left( {x,z,t} \right)}}{\partial z^{2}} + \frac{\partial^{2}{p_{1}\left( {x,z,t} \right)}}{\partial x^{2}} - {\frac{1}{c_{f}^{2}}\frac{\partial^{2}{p_{1}\left( {x,z,t} \right)}}{\partial t^{2}}}} = 0},} & (1)\end{matrix}$where p₁(x,z,t) is the pressure (N/m²), z is the spatial location (m)normal to the plate, x is spatial location along the plate (m), c_(f) isthe compressional wavespeed of the fluid (m/s), t is time (s), and thesubscript one denotes the area on the projector side of the testmaterial. The motion of the material is governed by the equation [2]

$\begin{matrix}{{{{\mu{\nabla^{2}u}} + {\left( {\lambda + \mu} \right){{\nabla\nabla} \cdot u}}} = {\rho\frac{\partial^{2}u}{\partial t^{2}}}},} & (2)\end{matrix}$where ρ is the density (kg/m³), λ and μ are the complex Lamé constants(N/m²), • denotes a vector dot product; u is the Cartesian coordinatedisplacement vector of the material. The acoustic pressure in the fluidon opposite the projector side of the test specimen is governed by thewave equation and is written in Cartesian coordinates as

$\begin{matrix}{{{\frac{\partial^{2}{p_{2}\left( {x,z,t} \right)}}{\partial z^{2}} + \frac{\partial^{2}{p_{2}\left( {x,z,t} \right)}}{\partial x^{2}} - {\frac{1}{c_{f}^{2}}\frac{\partial^{2}{p_{2}\left( {x,z,t} \right)}}{\partial t^{2}}}} = 0},} & (3)\end{matrix}$where p₂(x,z,t) is the pressure (N/m²) and the subscript two denotes thearea opposite the projector side of the test material. The interfacebetween the first fluid and solid surface of the material at z=bsatisfies the linear momentum equation, which is [3]

$\begin{matrix}{{{\rho_{f}\frac{\partial^{2}{u_{z}\left( {x,b,t} \right)}}{\partial t^{2}}} = {- \frac{\partial{p_{1}\left( {x,b,t} \right)}}{\partial z}}},} & (4)\end{matrix}$where ρ_(f) is the density of the fluid (kg/m³). The interface betweenthe second fluid and solid surface of the material at z=a also satisfiesthe linear momentum equation, and is written as

$\begin{matrix}{{\rho_{f}\frac{\partial^{2}{u_{z}\left( {x,a,t} \right)}}{\partial t^{2}}} = {- {\frac{\partial{p_{2}\left( {x,a,t} \right)}}{\partial z}.}}} & (5)\end{matrix}$The above five equations are the governing partial differentialequations of the insertion loss experiment.

Equations (1)-(3) are now transformed from partial differentialequations into ordinary differential equations and then into algebraicexpressions. The acoustic pressure in equation (1) is modeled as afunction at definite wavenumber and frequency asp ₁(x,z,t)=P ₁(z,k _(x),ω)exp(ik _(x) x)exp(iωt),  (6)where ω is frequency (rad/s), k_(x) is the spatial wavenumber in the xdirection (rad/m), and i is the square root of −1. The spatialwavenumber is given by

$\begin{matrix}{{k_{x} = {\frac{\omega}{c_{f}}{\sin(\theta)}}},} & (7)\end{matrix}$where θ is the angle of incidence (rad) of the incoming acoustic wavewith θ=0 corresponding to excitation normal to the sample (or broadsideexcitation). Inserting equation (6) into equation (1) and solving theresulting ordinary differential equation yieldsP ₁(z,k _(x),ω)=H(k _(x),ω)exp(iγz)+P _(S)(ω)exp(−iγz).  (8)In equation (8), the first term on the right hand side represents thereradiated (or reflected) pressure field and the second term representsthe applied incident pressure field (the forcing function) acting on thestructure. The term H(k_(x),ω) is the wave propagation coefficient ofthe reflected pressure field and the term P_(S)(ω) is the source (orexcitation) level. Additionally,

$\begin{matrix}{{\gamma = \sqrt{\left( \frac{\omega}{c_{f}} \right)^{2} - k_{x}^{2}}},} & (9)\end{matrix}$where γ is the wavenumber of the acoustic pressure in the fluid.

Equation (2) is manipulated by writing the Cartesian coordinatedisplacement vector u as

$\begin{matrix}{{u = \begin{Bmatrix}{u_{x}\left( {x,y,z,t} \right)} \\{u_{y}\left( {x,y,z,t} \right)} \\{u_{z}\left( {x,y,z,t} \right)}\end{Bmatrix}},} & (10)\end{matrix}$with y denoting the direction into the material in FIG. 2. The symbol ∇is the gradient vector differential operator written inthree-dimensional Cartesian coordinates as [4]

$\begin{matrix}{{\nabla{= {{\frac{\partial}{\partial x}i_{x}} + {\frac{\partial}{\partial y}i_{y}} + {\frac{\partial}{\partial z}i_{z}}}}},} & (11)\end{matrix}$with i_(x) denoting the unit vector in the x-direction, i_(y) denotingthe unit vector in the y-direction, and i_(z) denoting the unit vectorin the z-direction; ∇² is the three-dimensional Laplace operatoroperating on vector u as∇² u=∇ ² u _(x) i _(x)+∇² u _(y) i _(y)+∇² u _(z) i _(z)′  (12)with ∇² operating on scalar u as

$\begin{matrix}{{{\nabla^{2}u_{x,y,z}} = {{\nabla{\cdot {\nabla\; u_{x,y,z}}}} = {\frac{\partial^{2}u_{x,y,z}}{\partial x^{2}} + \frac{\partial^{2}u_{x,y,z}}{\partial y^{2}} + \frac{\partial^{2}u_{x,y,z}}{\partial z^{2}}}}};} & (13)\end{matrix}$and the term ∇•u is called the divergence and is equal to

$\begin{matrix}{{\nabla{\cdot u}} = {\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y} + {\frac{\partial u_{z}}{\partial z}.}}} & (14)\end{matrix}$

The displacement vector u is written asu=∇φ+∇×{right arrow over (ψ)},  (15)where φ is a dilatational scalar potential, × denotes a vector crossproduct, and {right arrow over (ψ)} is an equivoluminal vector potentialexpressed as

$\begin{matrix}{\overset{->}{\Psi} = {\begin{Bmatrix}{\Psi_{x}\left( {x,y,z,t} \right)} \\{\Psi_{y}\left( {x,y,z,t} \right)} \\{\Psi_{z}\left( {x,y,z,t} \right)}\end{Bmatrix}.}} & (16)\end{matrix}$The structural problem is formulated as a two-dimensional response (y≡0and ∂(·)/∂y≡0) problem. Expanding equation (15) and breaking thedisplacement vector into its individual nonzero terms yields

$\begin{matrix}{{{u_{x}\left( {x,z,t} \right)} = {\frac{\partial{\phi\left( {x,z,t} \right)}}{\partial x} - \frac{\partial{\psi_{y}\left( {x,z,t} \right)}}{\partial z}}}{and}} & (17) \\{{u_{z}\left( {x,z,t} \right)} = {\frac{\partial{\phi\left( {x,z,t} \right)}}{\partial z} + {\frac{\partial{\psi_{y}\left( {x,z,t} \right)}}{\partial x}.}}} & (18)\end{matrix}$

Equations (17) and (18) are next inserted into equation (2), whichresults in

$\begin{matrix}{{{c_{d}^{2}{\nabla^{2}{\phi\left( {x,z,t} \right)}}} = \frac{\partial^{2}{\phi\left( {x,z,t} \right)}}{\partial t^{2}}}{and}} & (19) \\{{c_{s}^{2}{\nabla^{2}{\psi_{y}\left( {x,z,t} \right)}}} = \frac{\partial^{2}{\psi_{y}\left( {x,z,t} \right)}}{\partial t^{2}}} & (20)\end{matrix}$where equation (19) corresponds to the dilatational component andequation (20) corresponds to the shear component of the displacementfield [5]. Correspondingly, the constants c_(d) and c_(s) are thecomplex dilatational and shear wave speeds, respectively, and aredetermined by

$\begin{matrix}{{c_{d} = \sqrt{\frac{\lambda + {2\mu}}{\rho}}}{and}} & (21) \\{c_{s} = {\sqrt{\frac{\mu}{\rho}}.}} & (22)\end{matrix}$The relationship of the Lamé constants to the compressional and shearmoduli is shown as

$\begin{matrix}{{\lambda = \frac{E\;\upsilon}{\left( {1 + \upsilon} \right)\left( {1 - {2\upsilon}} \right)}}{and}} & (23) \\{{\mu = {G = \frac{E}{2\left( {1 + \upsilon} \right)}}},} & (24)\end{matrix}$where E is the complex Young's (compressional) modulus (N/m²), G is thecomplex shear modulus (N/m²), and υ is the Poisson's ratio of thematerial (dimensionless).

The conditions of infinite length and steady-state response are nowimposed, allowing the scalar and vector potential to be written asφ(x,z,t)=Φ(z)exp(ik _(x) x)exp(iωt),  (25)andψ_(y)(x,z,t)=Ψ(z)exp(ik _(x) x)exp(iωt).  (26)Inserting equation (25) into equation (19) yields

$\begin{matrix}{{{\frac{\mathbb{d}^{2}{\Phi(z)}}{\mathbb{d}z^{2}} + {\alpha^{2}{\Phi(z)}}} = 0},{where}} & (27) \\{{\alpha = \sqrt{k_{d}^{2} - k_{x}^{2}}},{and}} & (28) \\{k_{d} = {\frac{\omega}{c_{d}}.}} & (29)\end{matrix}$Inserting equation (26) into equation (20) yields

$\begin{matrix}{{{\frac{\mathbb{d}^{2}{\Psi(z)}}{\mathbb{d}z^{2}} + {\beta^{2}{\Psi(z)}}} = 0},{where}} & (30) \\{{\beta = \sqrt{k_{s}^{2} - k_{x}^{2}}},{and}} & (31) \\{k_{s} = {\frac{\omega}{c_{s}}.}} & (32)\end{matrix}$

The solution to equation (27) isΦ(z)=A(k _(x)ω)exp(iαz)+B(k _(x),ω)exp(−iαz),  (33)and the solution to equation (30) isΨ(z)=C(k _(x),ω)exp(iβz)+D(k _(x),ω)exp(−iβz),  (34)where A(k_(x),ω), B(k_(x),ω), C(k_(x),ω), and D(k_(x),ω) are wavepropagation constants that are determined below. The displacements cannow be written as functions of the unknown constants using theexpressions in equations (17) and (18). They are

$\begin{matrix}{\begin{matrix}{{u_{z}\left( {x,z,t} \right)} = {{U_{z}\left( {k_{x},z,\omega} \right)}{\exp\left( {{\mathbb{i}}\; k_{x}x} \right)}{\exp\left( {{\mathbb{i}}\;\omega\; t} \right)}}} \\{= \left\{ {{{\mathbb{i}\alpha}\left\lbrack {{{A\left( {k_{x},\omega} \right)}{\exp\left( {{\mathbb{i}\alpha}\; z} \right)}} - {{B\left( {k_{x},\omega} \right)}{\exp\left( {{- {\mathbb{i}\alpha}}\; z} \right)}}} \right\rbrack} +} \right.} \\\left. {{\mathbb{i}}\;{k_{x}\left\lbrack {{{C\left( {k_{x},\omega} \right)}{\exp\left( {{\mathbb{i}\beta}\; z} \right)}} + {{D\left( {k_{x},\omega} \right)}{\exp\left( {{- {\mathbb{i}\beta}}\; z} \right)}}} \right\rbrack}} \right\} \\{{{\exp\left( {{\mathbb{i}}\; k_{x}x} \right)}{\exp\left( {{\mathbb{i}\omega}\; t} \right)}},}\end{matrix}{and}} & (35) \\\begin{matrix}{{u_{x}\left( {x,z,t} \right)} = {{U_{x}\left( {k_{x},z,\omega} \right)}{\exp\left( {{\mathbb{i}}\; k_{x}x} \right)}{\exp\left( {{\mathbb{i}\omega}\; t} \right)}}} \\{= \left\{ {{{\mathbb{i}}\;{k_{x}\left\lbrack {{{A\left( {k_{x},\omega} \right)}{\exp\left( {{\mathbb{i}\alpha}\; z} \right)}} + {{B\left( {k_{x},\omega} \right)}{\exp\left( {{- {\mathbb{i}\alpha}}\; z} \right)}}} \right\rbrack}} -} \right.} \\\left. {{\mathbb{i}\beta}\left\lbrack {{{C\left( {k_{x},\omega} \right)}{\exp\left( {{\mathbb{i}\beta}\; z} \right)}} - {{D\left( {k_{x},\omega} \right)}{\exp\left( {{- {\mathbb{i}\beta}}\; z} \right)}}} \right\rbrack} \right\} \\{{\exp\left( {{\mathbb{i}}\; k_{x}x} \right)}{{\exp\left( {{\mathbb{i}\omega}\; t} \right)}.}}\end{matrix} & (36)\end{matrix}$The normal stress the top of the plate (z=b) is equal to opposite thepressure in the fluid. This expression is

$\begin{matrix}{{{\tau_{zz}\left( {x,b,t} \right)} = {{{\left( {\lambda + {2\mu}} \right)\frac{\partial{u_{z}\left( {x,b,t} \right)}}{\partial z}} + {\lambda\frac{\partial{u_{x}\left( {x,b,t} \right)}}{\partial x}}} = {- {p_{1}\left( {x,b,t} \right)}}}},} & (37)\end{matrix}$and the tangential stress at the top of the plate is zero and thisequation is written as

$\begin{matrix}{{\tau_{zx}\left( {x,b,t} \right)} = {{\mu\left\lbrack {\frac{\partial{u_{x}\left( {x,b,t} \right)}}{\partial z} + \frac{\partial{u_{z}\left( {x,b,t} \right)}}{\partial x}} \right\rbrack} = 0.}} & (38)\end{matrix}$The normal stress the bottom of the plate (z=a) is equal to opposite thepressure in the fluid. This expression is

$\begin{matrix}{{{\tau_{zz}\left( {x,a,t} \right)} = {{{\left( {\lambda + {2\mu}} \right)\frac{\partial{u_{z}\left( {x,a,t} \right)}}{\partial z}} + {\lambda\frac{\partial{u_{x}\left( {x,a,t} \right)}}{\partial x}}} = {- {p_{2}\left( {x,a,t} \right)}}}},} & (39)\end{matrix}$and the tangential stress at the bottom of the plate is zero and thisequation is written as

$\begin{matrix}{{{\tau_{zx}\left( {x,a,t} \right)} = {{\mu\left\lbrack {\frac{\partial{u_{x}\left( {x,a,t} \right)}}{\partial z} + \frac{\partial{u_{z}\left( {x,a,t} \right)}}{\partial x}} \right\rbrack} = 0}},} & (40)\end{matrix}$where p₂(x,b,t) in equation (39) represents the radiated acousticpressure in the fluid load on the opposite side of the acousticprojector.

The acoustic pressure in equation (3) is modeled as a function atdefinite wavenumber and frequency asp ₂(x,z,t)=P ₂(z,k _(x),ω)exp(ik _(x) x)exp(iωt),  (41)Inserting equation (41) into equation (3) and solving the resultingordinary differential equation yieldsP ₂(z,k _(x),ω)=K(k _(x),ω)exp(−iγz),  (42)which is the outgoing (or transmitted) acoustic energy in the secondfluid. The term K(k_(x),ω) is the wave propagation coefficient of thetransmitted pressure field. Note that there is no incoming wave energyon this side of the test specimen and thus only one exponential term ispresent.

Assembling equations (1)-(42) and letting b=0 yields the four-by-foursystem of linear equations that model the system. They areAx=b,  (43)where the entries of equation (43) are

$\begin{matrix}{{A_{11s} = {{{- \alpha^{2}}\lambda} - {2\alpha^{2}\mu} - {\lambda\; k_{x}^{2}}}},} & (44) \\{{A_{11f} = \frac{\rho_{f}\omega^{2}\alpha}{\gamma}},} & (45) \\{{A_{11} = {A_{11s} + A_{11f}}},} & (46) \\{{A_{12} = {A_{11s} - A_{11f}}},} & (47) \\{{A_{13s} = {2k_{x}{\beta\mu}}},} & (48) \\{{A_{13f} = \frac{\rho_{f}\omega^{2}k_{x}}{\gamma}},} & (49) \\{{A_{13} = {{- A_{13s}} + A_{13f}}},} & (50) \\{{A_{14} = {A_{13s} + A_{13f}}},} & (51) \\{{A_{21} = {{- 2}\mu\; k_{x}\alpha}},} & (52) \\{{A_{22} = {- A_{21}}},} & (53) \\{{A_{23} = {{\mu\beta}^{2} - {\mu\; k_{x}^{2}}}},} & (54) \\{{A_{24} = A_{23}},} & (55) \\{{A_{31} = {\left( {A_{11s} - A_{11f}} \right){\exp\left( {{\mathbb{i}\alpha}\; a} \right)}}},} & (56) \\{{A_{32} = {\left( {A_{11s} + A_{11f}} \right){\exp\left( {{- {\mathbb{i}\alpha}}\; a} \right)}}},} & (57) \\{{A_{33} = {\left( {{- A_{13s}} - A_{13f}} \right){\exp\left( {{\mathbb{i}\beta}\; a} \right)}}},} & (58) \\{{A_{34} = {\left( {A_{13s} - A_{13f}} \right){\exp\left( {{- {\mathbb{i}\beta}}\; a} \right)}}},} & (59) \\{{A_{41} = {A_{21}{\exp\left( {{\mathbb{i}\alpha}\; a} \right)}}},} & (60) \\{{A_{42} = {{- A_{21}}{\exp\left( {{- {\mathbb{i}}}\;\alpha\; a} \right)}}},} & (61) \\{{A_{43} = {A_{23}{\exp\left( {{\mathbb{i}\beta}\; a} \right)}}},} & (62) \\{{A_{44} = {A_{23}{\exp\left( {{- {\mathbb{i}\beta}}\; a} \right)}}},} & (63) \\{{x_{11} = {A\left( {k_{x},\omega} \right)}},} & (64) \\{{x_{21} = {B\left( {k_{x},\omega} \right)}},} & (65) \\{{x_{31} = {C\left( {k_{x},\omega} \right)}},} & (66) \\{{x_{41} = {D\left( {k_{x},\omega} \right)}},} & (67) \\{{b_{11} = {{- 2}{P_{S}(\omega)}}},} & (68) \\{{b_{21} = 0},} & (69) \\{{b_{31} = 0},{and}} & (70) \\{b_{41} = 0.} & (71)\end{matrix}$It is noted that the subscript s corresponds to terms related to thestructure and the subscript f corresponds to terms related to the fluid.Using equations (43)-(71) the solution to the constants A(k_(x),ω),B(k_(x),ω), C(k_(x),ω) and D(k_(x),ω) can be calculated at each specificwavenumber and frequency. Written in transfer function form withreference to the source excitation level, they are

$\begin{matrix}{\;{{\frac{A\left( {k_{x},\omega} \right)}{P_{S}(\omega)} = {\left\{ {{4A_{13s}A_{22}{A_{23}\left\lbrack {1 - {{\cos({\beta\alpha})}{\exp\left( {{\mathbb{i}\alpha}\; a} \right)}}} \right\rbrack}} - {A\;{{\mathbb{i}}\left( {{A_{11}A_{23}^{2}} + {A_{13f}A_{22}A_{23}}} \right)}{\sin\left( {\beta\; a} \right)}{\exp\left( {{- {\mathbb{i}\alpha}}\; a} \right)}}} \right\}\Delta^{- 1}}},}} & (72) \\{\;{{\frac{B\left( {k_{x},\omega} \right)}{P_{S}(\omega)} = {\left\{ {{4A_{13s}A_{22}{A_{23}\left\lbrack {1 - {{\cos\left( {\beta\; a} \right)}{\exp\left( {{\mathbb{i}\alpha}\; a} \right)}}} \right\rbrack}} + {4{{\mathbb{i}}\left( {{A_{12}A_{23}^{2}} - {A_{13f}A_{22}A_{23}}} \right)}{\sin\left( {\beta\; a} \right)}{\exp\left( {{\mathbb{i}\alpha}\; a} \right)}}} \right\}\Delta^{- 1}}},}} & (73) \\{\;{{\frac{C\left( {k_{x},\omega} \right)}{P_{S}(\omega)} = {\left\{ {{4A_{11s}A_{22}{A_{23}\left\lbrack {{- 1} + {{\cos\left( {\alpha\; a} \right)}{\exp\left( {{- {\mathbb{i}\beta}}\; a} \right)}}} \right\rbrack}} - {4{{\mathbb{i}}\left( {{A_{13}A_{22}^{2}} + {A_{11f}A_{22}A_{23}}} \right)}{\sin\left( {\alpha\; a} \right)}{\exp\left( {{- {\mathbb{i}\beta}}\; a} \right)}}} \right\}\Delta^{- 1}}},{and}}} & (74) \\{\;{{\frac{D\left( {k_{x},\omega} \right)}{P_{S}(\omega)} = {\left\{ {{4A_{11s}A_{22}{A_{23}\left\lbrack {1 - {{\cos\left( {\alpha\; a} \right)}{\exp\left( {{\mathbb{i}}\;\beta\; a} \right)}}} \right\rbrack}} + {4{{\mathbb{i}}\left( {{A_{14}A_{22}^{2}} + {A_{11f}A_{22}A_{23}}} \right)}{\sin\left( {\alpha\; a} \right)}{\exp\left( {{\mathbb{i}\beta}\; a} \right)}}} \right\}\Delta^{- 1}}},{where}}} & (75) \\{{\Delta = {\Delta_{1} + \Delta_{2} + \Delta_{3} + \Delta_{4} + \Delta_{5}}},} & (76) \\{{\Delta_{1} = {{\exp\left( {{- {\mathbb{i}\alpha}}\; a} \right)}{\exp\left( {{\mathbb{i}\beta}\; a} \right)}\left( {{A_{11}A_{23}} + {A_{14}A_{22}}} \right)^{2}}},} & (77) \\{{\Delta_{2} = {{- {\exp\left( {{- {\mathbb{i}\alpha}}\; a} \right)}}{\exp\left( {{- {\mathbb{i}\beta}}\; a} \right)}\left( {{A_{11}A_{23}} + {A_{13}A_{22}}} \right)^{2}}},} & (78) \\{{\Delta_{3} = {{- {\exp\left( {{\mathbb{i}\alpha}\; a} \right)}}{\exp\left( {{\mathbb{i}}\;\beta\; a} \right)}\left( {{A_{12}A_{23}} - {A_{14}A_{22}}} \right)^{2}}},} & (79) \\{{\Delta_{4} = {{\exp\left( {{\mathbb{i}\alpha}\; a} \right)}{\exp\left( {{- {\mathbb{i}\beta}}\; a} \right)}\left( {{A_{12}A_{23}} - {A_{13}A_{22}}} \right)^{2}}},{and}} & (80) \\{\Delta_{S} = {{- 8}A_{11s}A_{13s}A_{22}{A_{23}.}}} & (81)\end{matrix}$

The transfer function between the wall motion in the z direction at z=aand the wall motion in the z direction at z=b (=0) is now written usingequations (35), (72), (73), (74), and (75). Additionally, the individualterms from the matrix A are inserted into the expression resulting in

$\begin{matrix}{\begin{matrix}{{T_{ba}\left( {k_{x},\omega} \right)} = \frac{U_{z}\left( {k_{x},b,\omega} \right)}{U_{z}\left( {k_{x},a,\omega} \right)}} \\{{= \frac{\begin{matrix}{{{\kappa_{1}\left( {k_{x},\omega} \right)}{\sin\left( {\alpha\; a} \right)}{\cos\left( {\beta\; a} \right)}} +} \\{\left\lbrack {{{\kappa_{2}\left( {k_{x},\omega} \right)}{\cos\left( {\alpha\; a} \right)}} + {{\kappa_{3}\left( {k_{x},\omega} \right)}{\sin\left( {\alpha\; a} \right)}}} \right\rbrack{\sin\left( {\beta\; a} \right)}}\end{matrix}}{{{\kappa_{1}\left( {k_{x},\omega} \right)}{\sin\left( {\alpha\; a} \right)}} + {{\kappa_{2}\left( {k_{x},\omega} \right)}{\sin\left( {\beta\; a} \right)}}}},}\end{matrix}{where}} & (82) \\{{{\kappa_{1}\left( {k_{x},\omega} \right)} = {{\mathbb{i}\gamma}\left( {4{\rho\beta\alpha}\; k_{x}^{2}\omega^{2}} \right)}},} & (83) \\{{{\kappa_{2}\left( {k_{x},\omega} \right)} = {{{\mathbb{i}\gamma}\left( {{\omega^{2}\rho} - {2\mu\; k_{x}^{2}}} \right)}\left( {\beta^{4} - k_{x}^{4}} \right)}},{and}} & (84) \\{{\kappa_{3}\left( {k_{x},\omega} \right)} = {{- {\alpha\rho}_{r}}{{\omega^{2}\left( {\beta^{4} + {2\beta^{2}k_{x}^{2}} + k_{x}^{4}} \right)}.}}} & (85)\end{matrix}$Further manipulation of equation (82) results in

$\begin{matrix}{\begin{matrix}{{T_{ba}\left( {k_{x},\omega} \right)} = \frac{U_{z}\left( {k_{x},b,\omega} \right)}{U_{z}\left( {k_{x},a,\omega} \right)}} \\{{= \frac{\begin{matrix}{{{\sin\left( {\alpha\; a} \right)}{\cos\left( {\beta\; a} \right)}} +} \\{\left\lbrack {{{M\left( {k_{x},\omega} \right)}{\cos\left( {\alpha\; a} \right)}} + {{N\left( {k_{x},\omega} \right)}{\sin\left( {\alpha\; a} \right)}}} \right\rbrack{\sin\left( {\beta\; a} \right)}}\end{matrix}}{{\sin\left( {\alpha\; a} \right)} + {{M\left( {k_{x},\omega} \right)}{\sin\left( {\beta\; a} \right)}}}},}\end{matrix}{where}} & (86) \\{{{M\left( {k_{x},\omega} \right)} = \frac{\kappa_{2}\left( {k_{x},\omega} \right)}{\kappa_{1}\left( {k_{x},\omega} \right)}},{and}} & (87) \\{{N\left( {k_{x},\omega} \right)} = {\frac{\kappa_{3}\left( {k_{x},\omega} \right)}{\kappa_{1}\left( {k_{x},\omega} \right)}.}} & (88)\end{matrix}$Equations (86), (87), and (88) are a mathematical model of the ratio ofwall motion of the test specimen. These equations are written so thatthe transfer function (or experimental data) is a function of materialproperties. They will be combined in such a manner that the materialproperties become functions of the experimental data. This process isexplained in the next section.

For completeness, it is noted that the reflected acoustic field on theprojector side is

$\begin{matrix}{{P_{R}\left( {k_{x},\omega} \right)} = {\left\lbrack {{\left( \frac{\omega^{2}\rho_{f}}{{\mathbb{i}}\;\gamma} \right){U_{z}\left( {k_{x},b,\omega} \right)}} + 1} \right\rbrack{{\exp\left( {{\mathbb{i}\gamma}\; z_{b}} \right)}.}}} & (89)\end{matrix}$where z_(b) is the position where the field is evaluated (m). The totalpressure field on the projector side is a sum of the reflected field andthe phase shifted source level written asP _(Total)(k _(x),ω)=P _(R)(k _(x),ω)+P _(S)(ω)exp(−iγz _(b)).  (90)The transmitted pressure field on the opposite side of the projector is

$\begin{matrix}{{P_{T}\left( {k_{x},\omega} \right)} = {\left\lbrack {\left( \frac{{- \omega^{2}}\rho_{f}}{\mathbb{i}\gamma} \right){U_{z}\left( {k_{x},a,\omega} \right)}} \right\rbrack{{\exp\left( {{- {\mathbb{i}\gamma}}\; z_{a}} \right)}.}}} & (91)\end{matrix}$where z_(a) is the position where the field is evaluated (m). Theinsertion loss is then calculated using

$\begin{matrix}{{{IL}\left( {k_{x},\omega} \right)} = {20\;{{\log_{10}\left\lbrack \frac{P_{S}(\omega)}{P_{T}\left( {k_{x},\omega} \right)} \right\rbrack}.}}} & (92)\end{matrix}$where IL(k_(x),ω) is in units of decibels. These measurements are notnecessary for the calculation of material properties according to theinvention. z_(a) and z_(b) are the positions of hydrophones 14 and 16.

Applicant's measurement method is a two step method. In the first step,projector 12 provides acoustic waves to the sample at zero wavenumber.In view of equation (7), this means that the projector is oriented toprovide acoustic waves at an angle θ of 0. In the second step, projector12 provides acoustic waves to the sample at a non-zero wavenumber. Thismeans that the projector is oriented to project acoustic waves at anyangle θ other than 0.

The first part of the measurement method involves insonifying twoseparate pieces of the material at zero wavenumber. The second piece ofmaterial is twice as thick as the first piece of material. For zerowavenumber, equation (82) reduces to

$\begin{matrix}{{{T_{ba}\left( {0,\omega} \right)} = {\frac{U_{z}\left( {0,b,\omega} \right)}{U_{z}\left( {0,a,\omega} \right)} = {{{\cos\left( {\alpha\; a} \right)} + {{\alpha\left\lbrack \frac{{\mathbb{i}\rho}_{f}c_{f}}{\omega\rho} \right\rbrack}{\sin\left( {\alpha\; a} \right)}}} = {T_{1}(\omega)}}}},} & (93)\end{matrix}$and, written to correspond to the to the test piece that is twice asthick, becomes

$\begin{matrix}{{T_{b\; 2a}\left( {0,\omega} \right)} = {\frac{U_{z}\left( {0,b,\omega} \right)}{U_{z}\left( {0,{2a},\omega} \right)} = {{{\cos\left( {2\alpha\; a} \right)} + {{\alpha\left\lbrack \frac{{\mathbb{i}\rho}_{f}c_{f}}{\omega\rho} \right\rbrack}{\sin\left( {2\alpha\; a} \right)}}} = {{T_{2}(\omega)}.}}}} & (94)\end{matrix}$where T₁(ω) and T₂(ω) are the transfer function data from theexperiment. It is noted, based on examination of equations (93) and(94), that no shear energy is excited in the structure when excitationis at zero wavenumber. Equations (93) and (94) can be combined andreduced using a double angle trigonometric expression to yield

$\begin{matrix}{{{\cos\left( {\alpha\; h} \right)} = {\frac{{T_{2}(\omega)} + 1}{2{T_{1}(\omega)}} = \phi}},} & (95)\end{matrix}$where φ is typically a complex valued number and h is the thickness ofthe first specimen (m). Equation (95) can be expanded into real andimaginary parts and solved, resulting in a value for α at everyfrequency in which a measurement is made. The solution to the real partof α is

$\begin{matrix}{{{Re}(\alpha)} = \left\{ {\begin{matrix}{{\frac{1}{2h}{Arc}\;{\cos(s)}} + {\frac{n\;\pi}{2h}{neven}}} \\{{\frac{1}{2h}{Arc}\;{\cos\left( {- s} \right)}} + {\frac{n\;\pi}{2h}{nodd}}}\end{matrix},{where}} \right.} & (96) \\{{s = {\left\lbrack {{Re}(\phi)} \right\rbrack^{2} + \left\lbrack {{Im}(\phi)} \right\rbrack^{2} - \sqrt{\left\{ {\left\lbrack {{Re}(\phi)} \right\rbrack^{2} + \left\lbrack {{Im}(\phi)} \right\rbrack^{2}} \right\}^{2} - \left\{ {{2\left\lbrack {{Re}(\phi)} \right\rbrack}^{2} - {2\left\lbrack {{Im}(\phi)} \right\rbrack}^{2} - 1} \right\}}}},} & (97)\end{matrix}$and n is a non-negative integer and the capital A denotes the principalvalue of the inverse cosine function. The value of n is determined fromthe function s, which is a periodically varying cosine function withrespect to frequency. At zero frequency, n is 0. Every time s cyclesthrough π radians (180 degrees), n is increased by 1. When the solutionto the real part of α is found, the solution to the imaginary part of αis then written as

$\begin{matrix}{{{Im}(\alpha)} = {\frac{1}{h}\log_{e}{\left\{ {\frac{{Re}(\phi)}{\cos\left\lbrack {{{Re}(\alpha)}h} \right\rbrack} - \frac{{Im}(\phi)}{\sin\left\lbrack {{{Re}(\alpha)}h} \right\rbrack}} \right\}.}}} & (98)\end{matrix}$The real and imaginary parts of a from equations (96) and (98)respectively are combined to yield the complex wavenumber. Because thismeasurement is made at zero wavenumber (k_(x)≡0), this is equal to thedilatational wavenumber. Thus, the dilatational wavespeed is equal to

$\begin{matrix}{c_{d} = {\frac{\omega}{\left\lbrack {{{Re}(\alpha)} + {{\mathbb{i}}\;{{Im}(\alpha)}}} \right\rbrack}.}} & (99)\end{matrix}$To solve for the shear wavespeed, the specimen must be excited at anonzero wavenumber. This is done in the next section.

The second part of the measurement method involves insonifying threeseparate pieces of the material at nonzero wavenumber. The second pieceof material is twice as thick as the first piece of material, and thethird piece of material is three times as thick as the first piece ofmaterial. For nonzero wavenumber, the equations corresponding to thethree pieces is

$\begin{matrix}{\begin{matrix}{{T_{ba}\left( {k_{x},\omega} \right)} = \frac{U_{z}\left( {k_{x},b,\omega} \right)}{U_{z}\left( {k_{x},a,\omega} \right)}} \\{= \frac{\begin{matrix}{{{\sin\left( {\alpha\; a} \right)}{\cos\left( {\beta\; a} \right)}} +} \\{\left\lbrack {{{M\left( {k_{x},\omega} \right)}{\cos\left( {\alpha\; a} \right)}} + {{N\left( {k_{x},\omega} \right)}{\sin\left( {\alpha\; a} \right)}}} \right\rbrack{\sin\left( {\beta\; a} \right)}}\end{matrix}}{{\sin\left( {\alpha\; a} \right)} + {{M\left( {k_{x},\omega} \right)}{\sin\left( {\beta\; a} \right)}}}} \\{= {R_{1}(\omega)}}\end{matrix},} & (100) \\{\begin{matrix}{{T_{b\; 2a}\left( {k_{x},\omega} \right)} = \frac{U_{z}\left( {k_{x},b,\omega} \right)}{U_{z}\left( {k_{x},{2a},\omega} \right)}} \\{= \frac{\begin{matrix}{{{\sin\left( {2\alpha\; a} \right)}{\cos\left( {2\beta\; a} \right)}} +} \\{\left\lbrack {{{M\left( {k_{x},\omega} \right)}{\cos\left( {2\alpha\; a} \right)}} + {{N\left( {k_{x},\omega} \right)}{\sin\left( {2\alpha\; a} \right)}}} \right\rbrack{\sin\left( {2\beta\; a} \right)}}\end{matrix}}{{\sin\left( {2\alpha\; a} \right)} + {{M\left( {k_{x},\omega} \right)}{\sin\left( {2\beta\; a} \right)}}}} \\{= {R_{2}(\omega)}}\end{matrix},} & (101) \\{\begin{matrix}{{T_{b\; 3a}\left( {k_{x},\omega} \right)} = \frac{U_{z}\left( {k_{x},b,\omega} \right)}{U_{z}\left( {k_{x},{3a},\omega} \right)}} \\{= \frac{\begin{matrix}{{{\sin\left( {3\alpha\; a} \right)}{\cos\left( {3\beta\; a} \right)}} +} \\{\left\lbrack {{{M\left( {k_{x},\omega} \right)}{\cos\left( {3\alpha\; a} \right)}} + {{N\left( {k_{x},\omega} \right)}{\sin\left( {3\alpha\; a} \right)}}} \right\rbrack{\sin\left( {3\beta\; a} \right)}}\end{matrix}}{{\sin\left( {3\alpha\; a} \right)} + {{M\left( {k_{x},\omega} \right)}{\sin\left( {3\beta\; a} \right)}}}} \\{= {R_{3}(\omega)}}\end{matrix},} & (102)\end{matrix}$It is noted that the α and β wavenumbers have different values whencompared to the previous section due to their modification by thenonzero spatial wavenumber k_(x). This dependency is shown in equations(28) and (31). Equations (100), (101), and (102) are now combined, theconstants M and N are condensed out, and the sine and cosine terms arereduced using multiple angle trigonometric expressions. Additionally, itis noted thatcos(βa)=cos(αa)  (103)is one of the solutions to the resulting expression and this term isfactored out because it is extraneous. This results inU(k _(x),ω)cos²(βh)+V(k _(x),ω)cos(βh)+W(k _(x),ω)=0,  (104)where the constants U, V, and W, are, written with the wavenumber andfrequency dependence suppressed, equal toU=4R ₁└4R ₂ cos²(αa)−2R ₃ cos(αa)−R ₂−1┘,  (105)V=2[−2R ₁ cos(αa)+R ₂+1I2R ₃ cos(αa)+1],  (106)andW=(R ₂+1)└−4R ₁ cos²(αa)+2 cos(αa)+R ₁ +R ₃┘.  (107)where α was determined with equation (28) using the values of c_(d)calculated in the previous section. Equation (104) can be solved as

$\begin{matrix}{{{\cos\left( {\beta\; h} \right)} = {\frac{{- V} + \sqrt{V^{2} - {4{UW}}}}{2U} = \varphi_{+}}},{and}} & (108) \\{{{\cos\left( {\beta\; h} \right)} = {\frac{{- V} - \sqrt{V^{2} - {4{UW}}}}{2U} = \varphi_{-}}},} & (109)\end{matrix}$where φ₊ and φ⁻ are typically a complex valued numbers. Two values of φare present but only one is the correct number. At zero (and very low)frequency, the φ value closest to unity is the correct one to use. Asfrequency increases, every time the angle of the discriminant inequation (108) passes through π radians, the value of φ changes fromequation (108) to equation (109) or vice versa. Once the correct valueof φ is known, equation (108) or (109) can be expanded into real andimaginary parts and solved, resulting in a value for β at everyfrequency in which a measurement is made. The solution to the real partof β is

$\begin{matrix}{{{Re}(\beta)} = \left\{ {\begin{matrix}{{\frac{1}{2h}{Arc}\;{\cos(r)}} + {\frac{m\;\pi}{2h}{meven}}} \\{{\frac{1}{2h}{Arc}\;{\cos\left( {- r} \right)}} + {\frac{m\;\pi}{2h}{modd}}}\end{matrix},{where}} \right.} & (110) \\{{r = {\left\lbrack {{Re}(\varphi)} \right\rbrack^{2} + \left\lbrack {{Im}(\varphi)} \right\rbrack^{2} - \sqrt{\left\{ {\left\lbrack {{Re}(\varphi)} \right\rbrack^{2} + \left\lbrack {{Im}(\varphi)} \right\rbrack^{2}} \right\}^{2} - \left\{ {{2\left\lbrack {{Re}(\varphi)} \right\rbrack}^{2} - {2\left\lbrack {{Im}(\varphi)} \right\rbrack}^{2} - 1} \right\}}}},} & (111)\end{matrix}$and m is a non-negative integer and the capital A denotes the principalvalue of the inverse cosine function. The value of m is determined fromthe function r, which is a periodically varying cosine function withrespect to frequency. At zero frequency, m is 0. Every time r cyclesthrough π radians (180 degrees), m is increased by 1. When the solutionto the real part of β is found, the solution to the imaginary part of βis then written as

$\begin{matrix}{{{Im}(\beta)} = {\frac{1}{h}\log_{e}{\left\{ {\frac{{Re}(\varphi)}{\cos\left\lbrack {{{Re}(\beta)}h} \right\rbrack} - \frac{{Im}(\varphi)}{\sin\left\lbrack {{{Re}(\beta)}h} \right\rbrack}} \right\}.}}} & (112)\end{matrix}$The real and imaginary parts of β from equations (110) and (112)respectively are combined to yield the complex wavenumber. Because thismeasurement is made at nonzero wavenumber, this has to be modified bythe spatial wavenumber k_(x) to calculate the shear wavenumber. Thisequation isk _(s)=√{square root over (β² +k _(x) ²)}.  (113)The shear wavespeed is then calculated using

$\begin{matrix}{c_{s} = {\frac{\omega}{k_{s}}.}} & (114)\end{matrix}$Once the dilatational and shear wavespeeds are known, the Lamé constantsor Young's modulus, shear modulus, and Poisson's ratio can also becalculated. A numerical example of all these calculations is includedbelow.

The above measurement method can be simulated by means of a numericalexample. Soft rubber-like material properties are used in thissimulation. The material has a Young's modulus E of{1e7(1−0.20i)[1+(1e−4)f]}N/m² where f is frequency in Hz, Poisson'sratio υ equal to 0.45 (dimensionless), and a density of ρ equal to 1200kg/m³. The base thickness of the material h is 0.01 m, the othertransfer functions (subscripts 2 and 3) are calculated using two andthree times this value. The water has a density ρ_(f) of 1025 kg/m³ anda compressional (acoustic) wave velocity of c_(f) of 1500 m/s. All otherparameters can be calculated from these values.

FIGS. 3A and 3B are plots of transfer function of normal wall motion atz=b divided by normal wall motion at z=a versus frequency at zerowavenumber (θ=0°). The x's correspond to h=0.01 m thickness and the +'scorrespond to h=0.02 m thickness. FIG. 3A is the magnitude, and FIG. 3Bis the phase angle. These functions are listed above as equations (93)and (94), respectively. FIG. 4 is a plot of the function s versusfrequency and corresponds to equation (97). The values of n in equation(96) can be determined from inspection of FIG. 4 and are listed in Table1, below.

TABLE 1 Minimum Maximum Frequency Frequency n (Hz) (Hz) 0 0 5660 1 566010000

FIGS. 5A and 5B are plots of the actual and estimated values ofwavenumber a versus frequency. FIG. 5A is the real part, and FIG. 5B isthe imaginary part. The actual values are shown with a solid line andthe estimated values are depicted with square markers. FIGS. 6A and 6Bare plots of the actual and estimated values of dilatational wavespeedversus frequency. FIG. 6A is the real part, and FIG. 6B is the imaginarypart. The actual values are shown with a solid line and the estimatedvalues are depicted with square markers.

FIGS. 7A and 7B are plots of transfer function of normal wall motion atz=b divided by normal wall motion at z=a versus frequency at wavenumberscorresponding to an insonifcation angle of 15 degrees (θ=15+). The x'scorrespond to h=0.01 m thickness, the +'s correspond to h=0.02 mthickness, and the o's correspond to h=0.03 m. FIG. 7A is the magnitude,and FIG. 7B is the phase angle. These functions are listed above asequations (100), (101), and (102), respectively. FIG. 8 is a plot of thefunction r (solid line with markers) and the angle of the discriminant(dashed line) versus frequency and corresponds to equation (111) and(108) respectively. Also included in this plot is the function rcalculated using φ, (equation 108) and .φ⁻ (equation 109) so that theinterchange relationship between these two functions and thediscriminant can be illustrated. The values of m in equation (110) canbe determined from inspection of FIG. 8 and are listed in Table 2,below.

TABLE 2 Minimum Maximum Frequency Frequency m (Hz) (Hz) 0 0 1460 1 14603110 2 3110 5000 3 5000 7120 4 7120 9500 5 9500 10000

FIGS. 9A and 9B are plots of the actual and estimated values ofwavenumber β versus frequency. FIG. 9A is the real part, and FIG. 9B isthe imaginary part. The actual values are shown with a solid line andthe estimated values are depicted with square markers. FIGS. 10A and 10Bare plots of the actual and estimated values of shear wavespeed versusfrequency. FIG. 10A is the real part, and FIG. 10B is the imaginarypart. The actual values are shown with a solid line and the estimatedvalues are depicted with square markers.

Finally, the material properties can be determined from the wavespeeds.The Lamé constants are calculated with equations (21) and (22) writtenasμ=ρc_(s) ²  (115)andλ=ρc _(d) ²−2ρc _(s) ².  (116)Alternatively, shear modulus, Poisson's ratio, and Young's modulus andcan be calculated using equations (23), (24), and (115) which results in

$\begin{matrix}{{{G \equiv \mu} = {\rho\; c_{s}^{2}}},} & (117) \\{{\upsilon = \frac{\lambda}{2\left( {\mu + \lambda} \right)}},{and}} & (118) \\{{E = \frac{2{\mu\left( {{2\mu} + {3\lambda}} \right)}}{2\left( {\mu + \lambda} \right)}},} & (119)\end{matrix}$respectively. FIGS. 11A and 11B are plots of the actual and estimatedvalues of Lamé constant μ versus frequency. FIG. 11A is the real part,and FIG. 11B is the imaginary part. The actual values are shown with asolid line and the estimated values are depicted with square markers.This corresponds to equation (115). FIGS. 12A and 12B are plots of theactual and estimated values of Lamé constant λ versus frequency. FIG.12A is the real part, and FIG. 12B is the imaginary part. The actualvalues are shown with a solid line and the estimated values are depictedwith square markers. This corresponds to equation (116). The shearmodulus G is identical to the Lamé constant μ and therefore is notplotted. Estimation of Poisson's ratio υ yields a value of 0.45(dimensionless). Because this is a constant with respect to frequency,it is not shown as a figure. FIGS. 13A and 13B are plots of the actualand estimated values of Young's modulus E versus frequency. FIG. 13A isthe real part, and FIG. 13B is the imaginary part. The actual values areshown with a solid line and the estimated values are depicted withsquare markers. This corresponds to equation (119).

In light of the above, it is therefore understood that within the scopeof the appended claims, the invention may be practiced otherwise than asspecifically described.

1. A method for calculating material properties of a material ofinterest comprising the steps of: determining a dilatational wavespeedby: conducting an insertion loss test of a first piece of the materialhaving a first thickness at zero wavenumber to obtain first transferfunction data; conducting an insertion loss test of a second piece ofthe material having a second thickness at zero wavenumber to obtainsecond transfer function data, wherein said second thickness is twicesaid first thickness; and calculating the dilatational wavespeed fromsaid first transfer function and said second transfer function;determining a shear wavespeed by: conducting an insertion loss test of afirst piece of the material having a first thickness at a non-zerowavenumber to obtain first shear transfer function data; conducting aninsertion loss test of a second piece of the material having a secondthickness at a non-zero wavenumber to obtain second shear transferfunction data, wherein said second thickness is twice said firstthickness; conducting an insertion loss test of a third piece of thematerial having a third thickness at zero wavenumber to obtain thirdshear transfer function data, wherein said third thickness is threetimes said first thickness; calculating the shear wavespeed from saidfirst shear transfer function, said second shear transfer function, saidthird shear transfer function and said dilatational wavespeed; andproviding said calculated shear wavespeed and said calculateddilatational wavespeed from said steps of calculating as the materialproperties.
 2. The method of claim 1 wherein said steps of conducting aninsertion loss test comprise: providing a sample of a material ofinterest having a first side and a second side; providing acousticenergy to said sample on a second side thereof at a selected frequencyat an angle selected by reference to the wavenumber; measuring a firstacceleration of said sample at said first side; measuring a secondacceleration of said sample at said second side; and computing atransfer function as a ratio of said second acceleration to said firstacceleration; and providing said computed transfer function as one ofthe material properties.
 3. The method of claim 2 wherein said firstthickness ranges from 10 mm to 100 mm.
 4. The method of claim 2 whereinsaid steps of measuring a first acceleration and measuring a secondacceleration are conducted by utilizing a laser velocimeter.
 5. Themethod of claim 2 wherein said steps of measuring a first accelerationand measuring a second acceleration are conducted by utilizing anaccelerometer.
 6. The method of claim 2 wherein said insertion losstests are conducted in a liquid filled housing.
 7. The method of claim 2wherein said insertion loss tests are conducted in a gas filled housing.8. The method of claim 2 further comprising the step of computing atleast one of Lamé constants, Young's modulus, Poisson's ratio, and theshear modulus for the material of interest using said dilatationalwavespeed and said shear wave speed.
 9. The method of claim 1 whereinsaid first thickness ranges from 10 mm to 100 mm.
 10. The method ofclaim 1 wherein said insertion loss tests are conducted in a liquidfilled housing.
 11. The method of claim 1 wherein said insertion losstests are conducted in a gas filled housing.
 12. The method of claim 1further comprising the step of computing at least one of Lamé constants,Young's modulus, Poisson's ratio, and the shear modulus for the materialof interest using said dilatational wavespeed and said shear wave speed.13. A method for obtaining the dilatational wavespeed of a materialcomprising the steps of: conducting an insertion loss test of a firstpiece of the material having a first thickness at zero wavenumber toobtain first transfer function data; conducting an insertion loss testof a second piece of the material having a first thickness at zerowavenumber to obtain second transfer function data; calculating thedilatational wavespeed from said first transfer function and said secondtransfer function; and providing the calculated dilatational wavespeedas one of the material properties.
 14. The method of claim 13 whereinsaid steps of conducting an insertion loss test comprise: providing asample of a material of interest having a first side and a second side;providing acoustic energy to said sample on a second side thereof at aselected frequency at an angle selected by reference to the wavenumber;measuring a first acceleration of said sample at said first side;measuring a second acceleration of said sample at said second side; andcomputing a transfer function as a ratio of said second acceleration tosaid first acceleration; and providing said computed transfer functionas one of the material properties.
 15. A method for calculating materialproperties of a material of interest comprising the steps of:determining a dilatational wavespeed by: conducting an insertion losstest of a first piece of the material having a first thickness at zerowavenumber to obtain first transfer function data; conducting aninsertion loss test of a second piece of the material having a secondthickness at zero wavenumber to obtain second transfer function data,wherein said second thickness is twice said first thickness; andcalculating the dilatational wavespeed from said first transfer functionand said second transfer function; determining a shear wavespeed by:conducting an insertion loss test of a first piece of the materialhaving a first thickness at a non-zero wavenumber to obtain first sheartransfer function data; conducting an insertion loss test of a secondpiece of the material having a second thickness at a non-zero wavenumberto obtain second shear transfer function data, wherein said secondthickness is twice said first thickness; conducting an insertion losstest of a third piece of the material having a third thickness at zerowavenumber to obtain third shear transfer function data, wherein saidthird thickness is three times said first thickness; calculating theshear wavespeed from said first shear transfer function, said secondshear transfer function, said third shear transfer function and saiddilatational wavespeed; and providing plots of said transfer functionsfrom said steps of conducting insertion loss tests.
 16. The method ofclaim 15 further comprising the steps of: computing at least one of Laméconstants and Young's modulus for the material of interest using saiddilatational wavespeed and said shear wavespeed; and providing a plot ofsaid computed one of Lamé constants and Young's modulus for the materialof interest.
 17. The method of claim 15 further comprising the step ofproviding a plot of said dilatational wavespeed versus frequency. 18.The method of claim 15 further comprising the step of providing a plotof said shear wavespeed versus frequency.